Romanovski polynomials, Gegenbauer connections, and $\mathrm{su}(1,1)$ ladder structures

Abstract

We study the monic Romanovski (pseudo-Jacobi) polynomials corresponding to the degree dependent parameters $β_K=-K$, $α_K=c/(K+1)$, with $K=n+\ell$. We write down, in explicit form, the parameter preserving first order lowering and raising relations at fixed $(α,β)$, with real proportionality constants. Combining them one recovers the standard three-term recurrence existing in any hypergeometric-type family, and iterating them one gets an ordered first order construction of $R_n^{α,β}$ starting from the unit constant polynomial. We also solve the connection problem with the $α=0$ family in a finite triangular form, identifying this last family with the Gegenbauer polynomials, and transferring the ladder relations to the $(n,\ell)$ lattice. After the substitution $x=\cotχ$, the in-level operators depend on $\ell$, but not on $K$. The resulting dressed functions support intrinsic lowest weight $\mathrm{su}(1,1)$ modules along the columns with $\ell$ fixed, while the circular row ($n=0$) requires an explicit boundary prescription and a rescaling.